A submodule over $A_0$ of $M_n$ can be written as the intersection with $M_n$ of the $A$-module it generates. Now try ascending chains.
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Wilberd van der KallenSep 18 '12 at 16:05

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The proof suggested by Wilberd shows in fact that if $G$ is a commutative group, $R$ is a commutative $G$-graded ring, and $M$ is a noetherian $G$-graded $R$-module, then every component of $M$ is a noetherian $R_0$-module.
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Fred RohrerSep 18 '12 at 21:03

2 Answers
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The answer is yes. Instead of showing that $M_n$ is finitely generated we may show it has the
property that any ascending sequence of $A_0$-submodules stabilizes. If $N$ is an $A_0$-submodule of $M_n$, consider $M_n\cap AN$. It is a sum of the $M_n\cap NA_i$. One sees it is
$N$ itself. So $N$ can be recovered from $AN$. Now if $N_1\subset N_2\dots$ is an ascending sequence of $A_0$-submodules of $M_n$, the ascending sequence $AN_1\subset AN_2\dots$ stabilizes, hence so do the $N_i=M_n\cap AN_i$.